Pure fluid logic circuitry for integrators and differentiators



Nov. 3, 1964 w. A. Boo-tHE 3,155,825

PURE FLUID LOGIC CIRCUITRY FOR INTEGRATORS AND DIFFERENTIATORS Filed Feb. 2l, 1965 2 Sheets-Sheet 1 Nov. 3,

W. A. BOOTHE PURE FLUID LOGIC CIRCUITRY FOR INTEGRATORS AND DIFFERENTIATORS Filed Feb. 2l, 1965 2 Sheets-Sheet 2 v 5;@ ...Q QQ

fr? Vent-orh//y//s f1. 500m@ United States Patent O 3,l55,tl25 PURE l'FlfJlUlD MlGl@ CllltCllllllY FR lbl'f- GRA'ERS AND Bllfthllllhlslll'lflll Willis A., liloothe, Scotia, NSY., assigner to General Electric Company, a corporation of New Yorlr Filed Felt. 2l, 1963, Ser. No. 260,2lll l5 Claims. (Si. 235-2tll) My invention relates to analog type fluid amplifier circuits, and in particular, to fluid amplifier circuits that perform mathematical integration and differentiation computation.

Fluid amplifiers are currently finding wide application in various fields and may be employed as digital and analog computing elements in fluid controlled systems and computer circuits and also as power devices to operate valves and the like. Fluid amplifiers feature inherent reliability and low cost due to their ease of fabrication form virtually any material that is nonporous and has structural integrity. As a result, they are ideal for applications where nuclear radiation, high temperature, vibration, and shock may be present. rEhe devices may be operated by employing either incompressible fluids such as liquids or compressible fluids such as gases, including air.

Fluid amplifiers fall into two basic categories, the digital type and the analog type. The analog type fluid amplifier is commonly referred to as the momentum exchange type wherein a power fluid jet is normally directed midway between two fluid receivers which may be spaced apart or adjoining each other, and is deflected relative to the receivers by an amount proportional to net sideways momenturn produced by control fluid jets directed laterally at the power jet from opposite sides thereof. My invention is directed to this latter type of fluid amplifier and particularly to the applications of this amplifier as a computing element for performing solutions of differential equations.

Therefore, one of the principal objects of my invention is to provide a fluid amplifier circuit for performing mathematical computations by calculus methods including integration and differentiation.

Another important object of my invention is to provide a new analog fluid amplifier circuit operable with a compressible or incompressible fluid wherein fluid pressure change at the fluid receivers is an integral or differential function of fluid pressure change at the input to fluid passages generating the control jets.

Briefly stated, and in accord with my invention, l provide an analog fluid amplifier adapted to generate a power jet of fluid which is controllably deflected relative to a pair of fluid receivers by control jets directed against the sides of the power jet. ln addition, l provide feedback fluid passages having input ends in fluid communication with the receivers and terminating in fl id flow restrictors adapted to generate feedback jets to further controllably deflect the power jet. A selected number of fluid flow restrictors for generating the control and feedback jets are disposed along opposite sides of the power jet in a manner whereby the feedback jets operate in both positive and negative feedback relation to the control jets. Suitable lluid flow impedances are disposed in the control and feedback fluid passages for producing desired fluid flow characteristics whereby selected fluid flow time constants are cancelled and the fluid pressure change at the receivers represents a calculus mathematical function of fluid pressure change at the input to the control fluid passages.

A fluid integrator circuit is obtained by providing two of the feedback fluid passages in communication with a common fluid receiver and arranging the two feedback fluid restrictors associated therewith in both positive and negative feedback relation to a control fluid restrictor. Each feedback fluid passage and control fluid passage includes respectively two and four predetermined fluid flow impedances.

A fluid differentiator circuit is obtained by providing one feedback fluid passage in communication with a fluid receiver and arranging the feedback fluid restrictor associated therewith in both positive and negative feedback relation to a pair of oppositely disposed control fluid restrictors wherein the two control fluid passages associated therewith have a common input connection. Each control fluid passage and feedback fluid passage includes respectively two and four predetermined fluid flow impedances.

T he features of my invention which l desire to protect herein are pointed out with particularity in the appended claims. rl`he invention itself, however, both as to its organization and method of operation, together with further objects and advantages thereof, may best be understood by reference to the following description taken in connection with the accompanying drawings, wherein like parts in each of the several figures are identified by the same reference character, and wherein:

FGURE l is a diagrammatic view in top plan illustrating an analog fluid integrator circuit constructed in accordance with my invention;

FIGURE 2 is a schematic representation of the fluid integrator circuit of FlGURE 1, shown partly in electrical circuit analogy;

FlGURlE 3 is a diagrammatic view in top plan of an analog fluid diiferentiator circuit constructed in accordance with my invention;

FGURE 4 is a schematic representation of FIGURE 3; and

FlGURE 5 is a diagrammatic view in top plan, partly broken away, of a fluid integrator circuit operable with both positive and negative differential fluid pressures.

ln the fluid amplifier circuit, diagrammatically illustrated in FIGURE l, there is shown a plurality of fluid amplifiers serially connected to form a high gain analog fluid amplifier circuit. Each individual fluid amplifier may comprise a defined fluid flow configuration contained in a separate plate or base member formed of any suitable nonporous structurally rigid material, such as metal, glass, plastic, or the like, but more preferably the seriallyconnected amplifiers are fabricated in a single plate. The fluid flow is conned within the defined configuration by means of a cover plate or flat cover member (not shown). Each fluid flow configuration comprises a plurality of flow paths or fluid passages, each being preferably rectangular in cross section, although other cross sections, such as circular, may be employed. The three serially-connected amplifiers shown in FIGURE l are each indicated as a whole by numerals l, 2, and 3, respectively, and enclosed within the boundary of the dashed lines. Each amplifier comprises a main or power fluid passage d terminating in a fluid flow restrictor forming a nozzle 5 adapted to generate a main or power jet of fluid issuing therefrom. The input to power fluid passage l is connected to a source of fluid flow to be controlled, indicated as Ps in FIGURE 2, by means of conduit 6 extending through the cover plate (not shown). This source of main or power fluid is maintained at a relatively constant and high fluid pressure and flow rate. The supply pressure of the power fluid may be equal in each of the three amplifier stages or vary in increasing magnitude with successive stages. In the second and third stage ampliflers 2 and 3, a pair of control fluid passages 7, S terminate respectively in a pair of oppositely disposed fluid flow restrictors forming control nozzles 9 and lll adapted to generate control jets of fluid issuing therefrom. The first stage amplifier 1 contains only one control fluid passage 7 terminating in control nozzle 9. Each of the control nozzles is disposed adjacent a power nozzle and substantially perpendicular to the power jet issuing therefrom whereby each control jet is directed against a side of a power jet in momentum exchange relation. The input to the control fiuid passage 7 in the first stage amplifier 1l is connected to a source of either control fluid flow, such as the output from another fluid amplifier, or pressure being sensed, such as from a pitot tube in an air stream. This source of control iiuid is indicated as P1 in FiGURE 2 and connected to passage 7 by means of conduit il extending through the cover plate (not shown). The source of control uid may be controlled as to liuid pressure and ow rate by independent means (not shown) and the pressure and liow rate of the control fluid is generally lower than that of the power fluid. The control iuid pressure may even be below ambient pressure. A pair of fluid receiving passages 12, 13 are disposed remote from power nozzle 5 and downstream thereto in a manner whereby in the absence `of any control jet the power jet is distributed equally between receivers l2 and ll. Vent passages lli, l5 are disposed adjacent reecivers l2 and 13 on opposite sides of the power jet. The vent passages extend through the cover plate and are in communication with the ambient atmosphere to relieve fluid pressure in the receivers resulting from unusual loading conditions. Fluid receivers 12, i3 may also be spaced apart, in which case a portion of the power jet in its noncontrolled state impinges between the receivers and passes through a Vented passage (not shown), the remaining portion of the power jet being equally divided between the receivers. Control passages 7 and 8 in the second and third stage fluid amplifiers 2, 3 have their inputs connected respectively to the outputs of receivers l2 and 13 of the preceding stage. The receivers in the third stage amplifier 3 are connected respectively to output fiuid passages 1d and l?, and the iiuid pressure at one of these receivers or the differential fluid pressure between the two receivers constitutes the output of this multistage fluid amplifier circuit.

The elements of the fluid amplifier circuit hereinabove described form a conventional analog iiuid amplifier circuit. A feature of my invention includes a pair of feedback fluid passages 13 and 19 having their inputs in fluid communication with receiver 1.3 of the third stage amplifier 3 by means of passage 17. Since feedback passage i3 passes across output passage 16, suitable bypass connections must be provided. Feedback passages 1u and 19 terminate respectively in a pair of oppositely disposed uid iiow restrictors that constitute feedback nozzles Zit and 21 adapted to generate feedback jets of fluid issuing therefrom. Feedback nozzles 2u and 21 are positioned adjacent the power nozzle 5 in the first stage amplifier in a manner whereby the feedback jet issuing from feedback nozzle 2li is in a first polarity feedback relation to the control jet issuing from control nozzle 9 in the first stage, and the feedback jet issuing from feedback nozzle 21 is in the opposite polarity feedback relation thereto. This bipolarity or positive and negative feedback relation may be demonstrated by tracing the fiow paths of the deiiected power jets through the circuit. Thus, assume the fiuid pressure of the control jet issuing from control nozzle 9 in the first stage is above ambient pressure, the power jet issuing from power nozzle S in the first stage is thereby controllably deflected whereby a greater fluid fiow passes Vinto receiver 13 than receiver 12 of the first stage as shown by the arrow in FIGURE 1. In like manner, the power jets Vin the second and third stage amplifiers are substantially deflected into receivers 12 and 13, respectively, since the outputs of the receivers of the first and second stage provide respectively the inputs to the control nozzles in the second and third stage amplifiers. As a consequence, the feedback jet issuing from feedback nozzle 2i? is in aiding or positive feedback relation to the control jet issuing from control nozzle 9 in the first stage and the feedback jet from nozzle 21 is in negative feedback relation thereto. lt should be appreciated that the control and feedback passages and their associated nozzles will be in the same relative position as illustrated in FIGURE 1 for a circuit including any odd number of fluid amplifiers serially connected. ln the case of a circuit comprising an even number of seriallyconnected amplifiers, the control nozzle in the first stage will be oppositely disposed in relation to the power jet from that illustrated in FIGURE 1. In an alternative case in which the fluid pressure of the control jet in the first stage is below ambient pressure, the control nozzle is also oppositely disposed from that illustrated in FIG- URE l.

ln general, an analog iiuid amplifier circuit suitable for performing mathematical computations comprises a high gain fluid operational amplifier. This operational amplifier includes a plurality of serially-connected fluid amplifiers to achieve the desired high gain since fluid amplifiers are inherently lower gain elements than analogous electronic amplifiers. This high gain requirement is important for the same reason that high gain operational amplifiers are employed in electronic computing circuits, the high gain providing outputs having a high degree of accuracy wherein loading effects are negligible.

FGURE 2 is a schematic represention of the iiuid amplifier circuit of FlGURE l wherein the fluid circuit elements of FIGURE 1 are represented, as far as possible, as analogous electric circuit elements. Thus, a fiow restriction, such as a nozzle, sharp-edged orice, or a capillary section, is represented as an electrical resistance. lf the fluid being employed is a compressible gas, a fixed volume may be employed and represented as a capacitance that is electrically grounded on one side. In like manner, a hydraulic accumulator when used with an incompressible fluid may also be represented as a capacitance that is grounded on one side.

The analysis of a fluid operational amplilier is similar in many respects to that of an electronic ope-rational amplifier. The simplest form of a fluid operational amplifier, not illustrated in any of the included iigures, consists of several analog fluid amplifiers connected in series to obtain high open loop gain, plus an input impedance ZI and a single feedback impedance ZF. ln addition, it is necessary to represent a control resistance Re and a feedback resistance Rf, these two resistances being respectively the inherent resistance of the control and feedback nozzle of the first stage fiuid amplifier. The operational amplifier equation for this basic operational amplifier can be derived in terms of the relationship of the change in output presure APO at a receiver in the nal stage in fiuid communication with the feedback passage to the change in input pressure AP, of the fluid supplied to the control nozzle in the first stage as equation:

This relationship of output pressure to input pressure is therefore substantially identical to the well-known relationship of output voltage VO, to input voltage, V1 of an electronic operational amplifier where Vo and V1 are substituted for APO and Al, respectively:

ZI (1C),

Korn and Korn in Electronic Analog Computers, 2nd

Edition, McGraw-Hill Book Company, pp. 1840. As therein set forth, the transfer function, namely the ratio of output to input, of a true or pure integrator takes the form of:

where R and C are the values of resistance and capacitance in the integrating circuit, S is the Laplace transform and K is a constant determined by the parameters of the circuit.

As explained by Korn and Korn, simple integrators approximate true integration for only a very short computing time and only under conditions of extremely high amplifier gain and almost prohibitively high values of impedance. The transfer function of a simple electronic integrator employing an operational, high-gain feedback amplifier having a gain G, is given by:

Xi-(l-G)RCS1 By comparison of Equation 2b and 2a, it is apparent that the system defined by the formed does not produce pure or true integra-tion whereas the one defined by the latter does. Equation 2b approximates 2a when both the gain, G, and the impedance value of RC are very lugh. These conditions, however, are undesirable; further, the approximation to pure or true integration is good only for a short integrating time period, thereby rendering the system insufficient for many required applications.

As set forth on pages 170-180 of the aboveaidentiiied text, electronic integrators having parallel feedback paths are well-known in the art, and oder a reasonable approximation of true or pure integration. in addition to a permissible reduction in the RC impedance values, the advantage is attained in the double feedback integrator that the computation time is not as severely restricted.

The present invention teaches the adaptation to fluid systems of the double feedback technique employed in electronic integrators. ln the pas-t, attempts to develop a double feedback fluid integrator performing pure or true integration have been unsuccessful since the requisite circuit components enabling the construction of an electronic double feedback integra-tor are not available as fluid equivalents. In particular the feedback loop in an electronic integrator incorporates a capacitor connected in a blocking-type configuration, namely, in series between the output and input termials of the international amplifier. The equivalent of such a blocking capacitor is not available in pneumatic technology.

By reference to FIGURE 1 of the application, there is 70 tor connected in a blocking conguration, but rather a capacitor C1 connected to ground from its junction with the resistor RF1, the latter being the electronic equivalent of restriction 25.

Accumulator 22 and restrictor 25 constitute the rst feedback loop of the operational iiuid amplifier integrators of FIGURES 1 and 2, the impedance of which may be equated to its transfer function, as follows:

ZF1+Rf1=A1+T1S (3) where the constant A1 equals the sum of the resistive values of the feedback loop, or: A1=(R1{RF1), and

where T1, the time constant of the feedback loop, is expressed by:

ln an identical fashion, the second feedback loop comprising the accumulator 23 and restrictor 2e may be analyzed by setting the impedance thereof, as represented in FIGURE 2, equal its transfer function:

ZF2+Rfz=A2(1i-T2S) (4) In accordance with the evaluation of the first feedback loop impedance equations', it follows in identical fashion that: A2=(Rf2-}RF2) and To determine the operating equations of the iiuid integrators of FIGURES 1 and 2, there is first noted that a basic operating requirement of yan analog fluid ampliiier is that the change in pressure operating on one side of a power jet must be equal to the change in pressure on the other side thereof. In terms of the pressure produced by the control jet from nozzle 9, identified asA the input flow P1, and by the feedback jets from nozzles 2d and 2l, identified as the feedback iiows P11 and Pfg, this requirement is expressed by the following basic equation:

AP1+APf1=APr2 (5) Proceeding in a standard derivation, Equation 5 solves to:

in the analysis of the first and second feedback loops and given in Equations 3 and 4 above, Equation 6 becomes:

Z1 A1 A2 1-|Ro Rn (1-l-T1S) Rm (l-l-T2S)] (7 lt is noted that the time constant T1 and T2 both appear with 1+ terms, occurring both in the numerator of the first term on the right hand side of the equals sign in Equation 7, and in the denominator of the second term.

The employment of double, or both positive and negative, feedback introduces both positive and negative factors in the second term, the substraction thereby effected being used to advantage in obtaining the integrating action. Specifically, by establishing the following reiationship of Values:

azar o, naze@ Ril RiB Ri?. RfZ Equation 7 solves to:

AP, 'tri-Tos APM:

And multiplying the first right hand term of Equation 9 by The second term on the right hand side of Equation l() has now been made similar to the transfer function of a pure integrator, as set forth in Equation 2a. In Equation l0 there nevertheless appear the impure factors of (l-l-TlS) and (l-l-TgS) in the numerator of the first right hand term. As discussed with regard to Equation la, these factors will prevent the fluid integrator, thus far set forth, from performing the pure or true integration desired.

In accordance with the invention, however, these detrimental factors are removed by properly defining and selecting, in type and magnitude, the components of the input impedance Z1 appearing in the denominator of the first right hand term of Equation 9 such that, as will be shown mathematically, the time constants T1 and T2 are eliminated.

In accordance with the invention, the technique for eliminating T1 and T2 is to design an input impedance Z1 having two time constants', T3 and T4. Further, the transfer function of the input and output impedances must contain these time constants as equivalent expressions or factors. Such an input impedance Z1 is shown in FIGURES l and 2. In the identical manner by which the first and second feedback loops were analyzed in Equations 3 and 4 above, the input impedance may be equated to its transfer impedance and analyzed as follei TazRoa, TFT T3 Substituting in Equation 10 the value of Z1 shown by Equation l1,

l APi (RriCi- RrzCz) S (12) It is apparent that the deleterious time constant factors of (l-l-T1S) and (l-i-TZS) are eliminated or cancelled by the time constant factors (l-l-TgS) and (l-l-T4S) by establishing the following equation:

To establish the equality expressed in Equation 13, it is suflicient that: T1=T3 and T2=T .1. Equatmg T1=T3 and T 2:5? .1, and substituting the corresponding values of T1, T2, T3, and T4 as calculated from Equations 3,

Apel:

4, and 1l results in the following criteria for the input impedance necessary to produce pure integration:

where C1 is greater than C2. In view of Equations 13, 14, and l5, Equation l2 reduces to:

Equationl is seen to be of the same form as Equation 2a and therefore to constitu-te a true or pure integrating function. Thus, in accordance with the invention, the deleterious time constant effects of prior vart fluid integrators resulting in non-linear or impure integration have been eliminated. The structure, as taught by the invention and as required for satisfying the conditions imposed in Equations I4 and l5 to achieve the pure integration shown in Equation 16 will now be discussed.

In FIGURE l, hydraulic accumulator Z2, which may comprise any of a number of conventional hydraulic accumulatore available, is disposed in `feedback passage IS and is represented as capacitor C1, the reactive component of the feedback impedance ZF1 in the first feedback loop in FIGURE 2. Hydraulic accumulator 23 is disposed in feedback passage 19 and represented as capacitor C2 of the feedback impedance Z132 in the second feedback loop in FIGURE 2. Two identical accumulators 24 are disposed in control passage 7 and represented as capacitors C3, the reactive components of the input impedance Z1 in FIGURE 2, Hydraulic accumulators are employed in the case of the fluid in the amplifier circuit comprising an incompressible fluid whereas fixed volumes are employed if the fluid is a compressible gas.

A flow restriction such as sharp-edged orifice 2S is disposed in feedback passage IS and is represented in FIC- URE 2 as resistor RF1, the resistive component of the feedback impedance ZF1 in the first feedback loop in FIGURE 2. In like manner, sharp-edged orice 26 in feedback passage I9 and sharp-edged orices 27 and 2S in control passage 7 are represented, respectively, in FIG- URE 2 as resistors RFE, R11, and R12.

In a manner similar to that employed in electronic operational amplifiers, the input-output relationship of the analog fluid amplifier may be inverted by merely interchanging the input and feedback networks. Thus, by interchanging the input impedance and feedback impedance of an integrator, a differentiator may be produced as illustrated in FIGURES 3 and 4. The change in output pressure APQ1 becomes proportional to the rate of change input pressure A131. The fluid differentiator also possesses the inherent characteristic of the electronic differentiator, that of being a definite noise amplier. However, an advantage of the fluid amplifier over the electronic amplifier is the absence of danger of damaging the elements due to overloads produced by noise peaks.

Referring particularly to FIGURE 3, there is shown a high gain analog fluid amplifier circuit similar to that illustrated in FIGURE l. The fundamental distinctions between the differentiator and integrator circuits occur in the feedback fluid passage and control fluid passage. Thus, the integrator in FIGURE l employs a pair of feedback fluid passages I8, I9 in fluid communication with one receiver I3 whereas the differentiator employs only one feedback fluid passage IS in fluid communication with receiver I3. Further, the feedback fluid passages in the integrator circuit are provided with two fluid impedances 22, 25 and 23, 26, respectively, in passages 18 and I9, while the differentiator circuit employs four impedances 29, 30, Sl, 32 per feedback passage. Finally, the integrator circuit employs one input control fluid passage 7 provided with four fluid impedances, orifices 27, Ztl and two identical accumulators 24 therein, whereas the differentiator circuit employs two control fluid passages '7, 33 having a common input connection 34 and having their output ends terminating at oppositely disposed nozzles 9 and 35 and the two control passages are provided with two inipedances 36, 37 and 38, 39, respectively. The single feedback nozzle 20 is operable in positive feedback relation to control nozzle 9 and in negative feedback relation to control nozzle 35 in a circuit comprising an odd number of serially connected amplifiers and having an input control fluid controllable in a pressure range above ambient.

FIGURE 4 is a schematic representation of the differentiator circuit illustrated in FIGURE 3. Fixed volumes 29 and 3l in feedback passage IS of FIGURE 3 are suitable for use with a compressible fluid, and in the case of an incompressible fluid, hydraulic accumulators, similar to that illustrated in FIGURE l, would be employed in place thereof. Fixed volumes 29 and 31 are identical and represented in FIGURE 4 as capacitors C3 grounded on one side, the reactive components of feedback impedance ZF. Flow restrictions such as capillary sections tl and 32 in feedback passage llS are represented, respectively, as resistors RF1, RFE, the resistive components of ZF. Capillary 36 and fixed volume 37 in input control passage 7 are indicated, respectively, as the resistive R11 and reactive C1 components of input impedance ZU. Capillary 3S and fixed volume 39 in input control passage 33 are indicated, respectively, as the resistive Rig and reactive C2 components of input impedance X12 in FIGURE 4. Sharp-edged orifices as well as nozzles may be substituted for these capillary sections to develop the appropriate resistance to the fluid flow. The output fluid pressure Fol is developed at receiver I3 of the last stage amplifier.

Following an essentially identical line of reasoning as applied to the analysis of a fluid integrator circuit performing pure integration, as set forth hereinbefore, the derived equation for the dilferentiator circuit of FIGURE 4 may be shown to be:

where RH=R2=RD RF1=RC, R=RC. The criteria for pure differentiation occurs for where C2 is greater than C1.

The integrator and the differentiator circuits illustrated, respectively, in FIGURES l and 3 are suitable for use with what may be described as a single polarity input signal, that is, the fluid pressure of the input control fluid is in general controllable in only a single range, such as above ambient pressure. This control fluid pressure may, on the other hand, be controllable in a range below ambient pressure, in which case, the power jets would be deflected in directions opposite to that obtained by the above-ambient pressure case. The use of below-ambient pressure control fluid which may be described as a negative polarity input signal, would necessitate disposing the input control nozzle 9 in FIGURE 1 on the opposite side of the power jet. In the negative polarity input signal embodiment for the differentiator circuit in FIGURE 3,. the feedback nozzle Ztl would be disposed on the opposite side of the power jet from that illustrated.

Bi-polarity input signals may be employed and bi-polarity output pressure signals may be obtained with the circuit of FIGURE 5. FIGURE illustrates an integrator circuit, similar to that shown in FIGURE L.,

but including two separate control fluid passages provided with their corresponding independent control fluid supply sources and two pairs of feedback fluid passages, each pair in fluid communication `with a separate fluid receiver. The circuit illustrated in FIGURE 5 is partly broken away, omitting the fluid amplifiers following the rst stage which are similar to those disclosed in FIGURE l. The circuit of FIGURE 5 permits the use of a bi-polarity input control fluid wherein the input signal represents a differential pressure existing between two input control fluids. The nozzles terminating at the output end of the control i'iuid and feedback fluid passages are arranged adjacent power nozzle 5 and along two opposite sides of the power jet in a manner whereby control nozzle 9 and nozzles di), Ztl, associated respectively with a first of each pair of feedback fluid passages 4I and 1S, are disposed on a first side of the power jet and a second control nozzle 42 and nozzles 43, 2l, associated respectively with a second pair of feedback passages 44 and I9, are disposed on the opposite side of the power jet. The feedback jet issuing from nozzle 2li is in positive feedback relation to the control jet issuing from nozzle 9 and the feedback iet issuing from nozzle 4l! is in negative feedback relation thereto in the case of an odd number of fluid amplifiers being employed. In like manner, the feedback jet issuing from nozzle i3 is in positive feedback relation to the control jet issuing from nozzle 42 and the feedback jet issuing from nozzle 21 is in negative feedback relation thereto. The fluid impedances in passages I8 and 44 are equal and determined by the criteria established for the impedances in passage I8 of FIGURE l. Correspondingly, the fluid im edances in passages I9 and 4l are equal and determined by the criteria established for the impedances in passage I9 of FIGURE l. The fluid impedances in the second input control fluid passage 45 are equal to the iinpedances in the rst input control fluid passage 7 and also determined as for passage 7' in FIGURE l. The output of the integrator circuit is a differential fluid pressure between the two receivers l2, I3 in the last stage amplifier and represents the integral of the differential pressure between the two input signals. The output may thus be of bi-polarity since the fluid pressure in output passage le may be greater or less than the fluid pressure in output passage I7 as determined by the fluid pressure of the two control fluid inputs to the first stage.

The dilferentiator circuit illustrated in FIGURE 3 may be modified to a bi-polarity input and output circuit in a manner similarly to that illustrated for the integrator circuit. Thus, a second pair of control fluid passages having their input ends connected in common to a second input control signal and their output ends terminating in nozzles oppositely disposed and adjacent nozzles 9 and 35 of the first stage, and a second feedback fluid passage having its input end in fluid communication with output passage 16 and its output end terminating in a nozzle disposed on the opposite side of the power jet from nozzle Ztl, provide a bi-polarity ditferentiator circuit. The fluid impedances in both feedback passages would be equal and the fluid impedances in the second pair of input control passages would be equal to those in the first pair.

Having described single and bi-polarity embodiments of an analog uid integrator and dilierentiator circuit, it is believed obvious that modifications and variations `of my invention are possible in the light of the above teachings. Thus, the single-polarity integrator circuit illustrated in FIGURE 1 may develop a lai-polarity output by providing a constant pressure bias jet oppositely disposed to the control jet in the first stage and utilizing a differential output between receivers I2 and 13 in the 'nal stage. Further, the single-input integrator and differentiator circuits illustrated schematically in FIG- URES 2 and 4, respectively, may each develop a bipolarity or differential output by providing feedback passages in communication with receiver l2 and flow impedances which are identical to those in the other feedback passages to form symmetrical feedbacks. ln like manner, a differential input as illustrated in FlGURE 5 and single output may be provided for both the integrator and differentiator circuits by providing symmetrical input control fluid passages which are supplied from a second source of control fluid. lt is, therefore, to be understood that changes may be made in the particular embodiments of my invention described which are within the full intended scope of the invention and defined by the following claims.

What I claim as new and desire tosecure by Letters Patent of the United States is:

l. An analog fluid amplifier circuit for obtaining calculus mathematical functions comprising first means for generating a main fluid jet to be controlled,

fluid receiver means downstream from said first means for receiving the main jet,

second means for establishing a control fluid flow and generating a control fluid jet tot controllably deflect said main jet, said second means including a first plurality of circuit components,

third means in communication with said receiver means for establishing a feed-back flow and generating a feedback fluid jet to further controllably deflect said main jet, said third means including a second plurality of circuit component-s establishing fluid flow characteristics essential to the generation of said calculus mathematical functions,

said first and said second pluralities of circuit components including resistive and reactive elements defining in each of said first and second means transfer impedance functions containing equivalent time constant factors, and

means combining said control and feedback fluid flows for effecting cancellation of said equivalent time constant factors for rendering said analog fluid amplifier circuit operable to produce pure calculus mathematical ftmctions.

2. The combination of an analog fluid amplifier comprising first means terminating in a first fluid flow res-trictor for generating a jet of main fluid to be controlled, a pair of fluid receiver means downstream from said first restrictor for receiving fluid from the main jet, and control means terminating in a second fluid flow restrictor for establishing a control fluid flow and generating jets of control fluid to controllably deflect said main jet relative to said receiver means, with feedback means terminating in a `third fluid flow restrictor for establishing ya feedback fluid flow and generating jets of feedback fluid to further controllably deflect said main jet relative to said receiver means, input to said feedback means in communication with said receiver means, at least one each of said second and third restrictors disposed -along a first side of said main jet and at least one other restrictor from a group including said second and third restrictors disposed on the opposite side of said main jet whereby at least three restrictors are intermediate said first restrictor and said receiver means, and

said control means and said feedback means including,

respectively, first and second pluralities of circuit components, each of said pluralities including resistive and reactive elements defining in each of said control and feedback means transfer impedance functions containing equivalent time constant factors, and

means combining said control and feedback fluid flows for effecting cancellation of said equivalent time constant factors for rendering said analog fluid amplifier circuit operable to produce pure calculus mathematic cal functions.

3. The combination set forth in claim 2 wherein said l2 impedance means comprise fixed volumes and sharpedged orifices.

4. The combination set forth in claim 2 wherein said impedance means comprise fixed volumes and capillary sections.

5. An yanalog fluid integrator circuit comprising first means including a first fluid flow restrictor for generating a jet of power fluid to be controlled, a pair of fluid receiver means downstream from said first fluid flow restrictor for receiving the power jet,

control means including a second fluid flow restrictor for establishing a control fluid flow and generating at least one jet of control fluid to controllably deflect said power jet, feedback means including a third fluid flow restrictor for establishing a feedback fluid flow and generating at least one pair of jets of feedback'fluid to further controllably deflect said power jet, each pair of feedback means in communication with one of said receiver means, at least one said second restrictor and one said third restrictor disposed along a first side of said power jet, at least one said third restrictor disposed on the opposite side of said power jet, said second and third restricto-rs disposed adjacent said first restrictor, and said control means and said feedback means including, respectively, first and second pluralities of circuit components, each of said pluralities including resistive and reactive elements and defining in each of said control and feedback means transfer impedance functions containing equivalent time constant factors, and

means combining said control and feedback fluid flows for effecting cancellation of said equivalent time constant factors for rendering said analog fluid amplifier circuit operable to produce pure calculus mathematical functions.

6. A single-side analog fluid integrator circuit comprisa first fluid passage terminating in a first nozzle for generating a power jet of fluid to be controlled,

a pair of fluid receiver means downstream from said first nozzle for receiving fluid from the power jet,

a second fluid passage terminating in a second nozzle for generating a control jet of fluid which by momentum exchange controllably deflects said power jet relative to said receiver means,

a pair of third fluid passages each terminating in a third nozzle for generating a pair of feedback jets of fluid which by momentum exchange further controllably deflect said power jet relative to said receiver means, said pair of third passages in communication with one receiver means, said second nozzle and a first of said third nozzles disposed along a first side of said power jet, the first of said third nozzles operable in a predetermined polarity feedback relation to said second nozzle, the rem ining third nozzle disposed on the opposite side of said power jet and operable in opposite polarity feedback relation to said second nozzle, said second and third nozzles disposed adjacent said first nozzle, and

fluid flow impedance means disposed in said second and third passages for producing predetermined fluid flow characteristics therein whereby the fluid pressure change at the receiver means in communication with said third passages represents a mathematical integral function of fluid pressure change at the input to said second passage, said impedance means comprising four impedances in the second fluid passage and two impedances in each of the third fluid passages, said impedance means being interrelated in a predetermined manner whereby selected fluid flow time constants are canceled to thereby produce pure integral functions.

7. A push-pull analog fluid integrator circuit comprisa first fluid passage terminating in a first nozzle for generating a power jet of fluid to be controlled,

a pair of fluid receiver means downstream from said first nozzle for receiving fluid from the power jet,

a pair of second fluid passages each terminating in a second nozzle for generating a pair of control jets of fluid to controllably deflect said power jet relative to said receiver means, said second nozzles disposed adjacent said first nozzle and on opposite sides of said power jet,

two pairs of third fluid passages each terminating in a third nozzle disposed adjacent said second nozzles for generating two pairs of feedback jets of fluid to further controllably deflect said power jet relative to said receiver means, the first and second pair of third passages in communication respectively with the first and second of said receiver means, a first of said second nozzles and a first nozzle from each pair of third nozzles disposed along a first side of said power jet, the second of said second nozzles and the second nozzle from each pair of third nozzles disposed along the opposite side of each power jet whereby each pair of third nozzles is operable in both positive and negative feedback relation to a selected second nozzle, and

fluid flow impedance means disposed in said second and third passages for producing predetermined fluid flow characteristics therein whereby the differential fluid pressure change between said pair of receiver means represents a mathematical integral function of the differential fluid pressure change between inputs to said pair of second passages, said impedance means being interrelated in a predetermined manner whereby selected fluid flow time constants are canceled to thereby produce pure integral functions.

8. ln a single-side analog fluid integrator circuit adapted to operate with a compressible fluid comprising a first fluid passage terminating in a first nozzle adapted to generate a power jet of compressible fluid to be controlled,

a pair of fluid receiver means downstream from said first nozzle for receiving fluid from the power jet,

a second fluid passage terminating in a second nozzle adapted to generate a control jet of compressible fluid directed against a side of said power jet to controllably deflect said power jet relative to said receiver means,

a pair of third fluid passages each terminating in a third nozzle adapted to generate a pair of feedback jets of compressible fluid to further controllably deflect said power jet relative to said receiver means, said pair of third passages in communication with one receiver means, said second nozzle and a first of said third nozzles disposed along a first side of said power jet, the first of said third nozzles operable in a predetermined polarity feedback relation to said second nozzle, the remaining third nozzle disposed on the opposite side of said power jet and operable in opposite polarity feedback relation to said second nozzle, said second and third nozzles disposed adjacent said first nozzle,

a pair of vent passages opening at opposite sides of said power jet adjacent said receiver means, and

predetermined fluid flow impedances comprising fixed volumes having inlet and outlet passages axially displaced from each other and sharp-edged orifices disposed in said second and third passages for producing desired compressible fluid flow characteristics whereby selected fluid flow time constants are cancelled and the fluid pressure change at the receiver means in communication with said third passages represents a mathematical pure integral function of fluid pressure change at the input to said second passage.

9. The combination set forth in claim S wherein said a. predetermined fluid flow impedances comprise two fixed volumes and two flow restrictions in said second passage, and one fixed volume and one flow restriction in each said third passage.

l0. in a single-side analog fluid integrator circuit adapted to operate with an incompressible fluid comprising a first fluid passage terminating in a first nozzle adapted to generate a power jet of incompressible fluid to be controlled,

a pair of fluid receiver means downstream from said first nozzle for receiving fluid from the power jet,

a second fluid passage terminating in a second nozzle adapted to generate a control jet of incompressible fluid directed against a side of said power jet to controllably deflect said power jet relative to said receiver means,

a pair of third fluid passages each terminating in a third nozzle adapted to generate a pair of feedback jets of incompressible fluid to further controllably deflect said power jet relative to said receiver means, said pair of third passages in communication with one receiver means, said second nozzle and a first of said third nozzles disposed along a first side of said power jet, the first of said third nozzles operable in a preeterrnined polarity feedback relation to said second nozzle, the remaining third nozzle disposed on the opposite side of said power jet and operable in opposite polarity feedback relation to said second nozzle, said second and third nozzles disposed adjacent said first nozzle,

a pair of vent passages opening at opposite sides of said power jet adjacent said receiver means, and

said second and third passages including, respectively,

two and one capillary sections forming predetermined fluid flow impedances for producing desired incompressible fluid flow characteristics whereby selected fluid flow time constants introduced by reason of the impedances in third fluid passage are cancelled and the fluid pressure change at the receiver means in communication with said third passages represents a mathematical pure integral function of fluid pressure change at the input to said second passage.

1l. An analog fluid diiferentiator circuit comprising first means including a first fluid flow restrictor for generating a jet of power uid to be controlled,

fluid receiver means downstream from said first fluid flow restrictor for receiving the power jet,

control means including a second fluid flow restrictor for generating at least one pair of jets of control fluid to controllably deflect said power jet, each pair of said control means having a common connection at the input end thereof,

feedback means including a third fluid flow restrictor for generating at least one jet of feedback fluid to further controllably deflect said power jet, each said feedback means in communication with a receiver means, at least one said second restrictor and one said third restrictor disposed along a first side of said power jet, at least one said second restrictor disposed on the opposite side of said power jet, said second and third restrictors disposed adjacent said dist restrictor, and

fluid flow impedance means in communication with said control and feedback means for producing selected duid flow characteristics therein whereby the relation of fluid pressure change at said receiver means to fluid pressure change at the input to said control means represents a mathematical differential function, said impedance means being interrelated in a particular manner whereby selected fluid flow time constants are canceled to thereby produce a pure differential function.

l2. A single-side analog fluid differentiator circuit cornprising a first Huid passage terminating in a first nozzle for generating a power jet of fluid to be controlled,

a pair of fluid receiver means downstream from said first nozzle for receiving fluid from the power jet,

a pair of second huid passages each terminating in a second nozzle for generating a pair of control jets of fluid which by momentum exchange controllably deiiect said power jet relative to said receiver means, said second passages having a common connection at their input end,

a third uid passage terminating in a third nozzle for generating a feedback jet of tiuid which by momentum exchange further controllably deiiects said power jet relative to said receiver means, said third liti characteristics therein whereby the dilferential iiuid ressure change between said pair of receiver means represents a mathematical differential function of the differential fluid pressure change between inputs to said two pairs of second passages, said impedance means being interrelated in a particular manner whereby selected iiuid flow time constants are canceled to thereby produce a pure differential function.

14. In a single-side analog iiuid differentiator circuit adapted to operate with a compressible iiuid comprising a first fluid passage terminating in a first nozzle adapted to generate a power jet of compressible fluid to be controlled,

a pair of fluid receiver means downstream from said passage in communication with one receiver means, first nozzle for receiving iiuid from the power jet, said third nozzle and a first of said second nozzles a pair of second fluid passages each terminating in a disposed along a first side of said power jet, said second nozzle adapted to generate a pair of control third nozzle operable in a predetermined polarity jets of compressible fluid directed against opposite feedback relation to the first of said second nozzles, sides of said power jet to controllably deflect said the remaining second nozzle disposed on the oppopower jet relative to said receiver means, said second Site side of said power jet whereby said third nozzle passages having a common connection at their input is operable in opposite polarity feedback relation end, thereto, said second and third nozzles disposed adthird fluid passage terminating in a third nozzle jacent said first nozzle, and adapted to generate a feedback jet of compressible fluid flow impedance means in communication with fluid to further controllably deflect said power jet said second and third passages for producing prederelative to said receiver means, said third passage in termined fluid fiow characteristics therein whereby communication with one receiver means, said third the fluid pressure change at the receiver means in nozzle and a first of said second nozzles disposed communication with said third passage represents a along a first side of said power jet, said third nozzle mathematical differential function of fiuid pressure operable in a predetermined polarity feedback relachange at the input to said second passages, said imtion to the first of said second nozzles, the remainpedance means being interrelated in aparticular maning second nozzle disposed on the opposite side of ner whereby selected fluid flow time constants are said power jet whereby said third nozzle is operable canceled to thereby produce a pure differential funcin opposite polarity feedback relation thereto, said tion. second and third nozzles disposed adjacent said first nozzle, a pair of vent passages opening at opposite sides of said power jet adjacent said receiver means, and predetermined fluid iiow impedances comprising fixed volumes having inlet and outlet passages axially displaced from each other and sharp-edged orifices disposed in said second and third passages for produc- 13. A push-pull analog fluid diiferentiator circuit comprising a first fluid passage terminating in a first nozzle for generating a power jet of uid to be controlled, a pair of fluid receiver means downstream from said first nozzle for receiving iiuid from the power jet, two pairs of second fluid passages each terminating in a second nozzle for generating two pairs of control jets of uid to controllably deflect said power jet relative to said receiver means, each pair of said second passages connected together at the input end thereof whereby a first and second control fluid may be supplied respectively to the first and second ing desired compressible iiuid flow characteristics whereby selected fluid flow time constants are canceled and the fluid pressure change at the receiver means in communication with said third passage represents a mathematicaly differential function of fluid pressure change at the input to said second passages.

15. The combination set forth in claim 14 wherein said predetermined iluid iow impedances comprise two fixed volumes and two'ow restrictions in said third passage, and one fixed volume and one flow restriction in each said second passage.

References Cited in the file of this patent to said receiver means, the first and second of said UNITED STATES PATENTS third passages in communication respectively with the first and second of said receiver means, a first of 21g said third nozzles and a first nozzle from each pair r of said second nozzles disposed along a first side of said power jet, the second of said third nozzles and FOREIGN PATENTS posed along the opposite side of said power jet whereby each third nozzle is operable in both posi- OTHER REFERENCES tive and negative relation to the second nozzles, and An Analog Pure Fluid Amplifier, E. M. Dexter, apfluid flow impedance means disposed in said second and pearing in pamphlet by American Society of Mechanical third passages for producing predetermined fluid flow Engineers, New York, NY., Nov. 28, 1962, pages 41-49. 

1. AN ANALOG FLUID AMPLIFIER CIRCUIT FOR OBTAINING CALCULUS MATHEMATICAL FUNCTIONS COMPRISING FIRST MEANS FOR GENERATING A MAIN FLUID JET TO BE CONTROLLED, FLUID RECEIVER MEANS DOWNSTREAM FROM SAID FIRST MEANS FOR RECEIVING THE MAIN JET, SECOND MEANS FOR ESTABLISHING A CONTROL FLUID FLOW AND GENERATING A CONTROL FLUID JET TO CONTROLLABLY DEFLECT SAID MAIN JET, SAID SECOND MEANS INCLUDING A FIRST PLURALITY OF CIRCUIT COMPONENTS, THIRD MEANS IN COMMUNICATION WITH SAID RECEIVER MEANS FOR ESTABLISHING A FEED-BACK FLOW AND GENERATING A FEEDBACK FLUID JET TO FURTHER CONTROLLABLY DEFLECT SAID MAIN JET, SAID THIRD MEANS INCLUDING A SECOND PLU- 